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Wireless Network Pricing Chapter 5: Monopoly and Price Discriminations

Jianwei Huang & Lin Gao

Network Communications and Economics Lab (NCEL) Information Engineering Department The Chinese University of Hong Kong

Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 25, 2018 1 / 39

The Book

E-Book freely downloadable from NCEL website: http: //ncel.ie.cuhk.edu.hk/content/wireless-network-pricing Physical book available for purchase from Morgan & Claypool (http://goo.gl/JFGlai) and Amazon (http://goo.gl/JQKaEq)

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Chapter 5: Monopoly and Price Discriminations

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Focus of This Chapter

Key Focus: Profit maximization in a monopoly market

◮ One service provider (monopolist) dominates the market

Theoretic Approach: Price Theory

◮ The study of how prices are decided and how they go up and down

because of economic forces such as changes in supply and demand

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Price Theory

Part I: Monopoly Pricing

◮ The service provider charges a single optimized price to all consumers.

Part II: Price Discrimination

◮ The service provider charges different prices for different units of

products or to different consumers.

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Section 5.1 Theory: Monopoly Pricing

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What is Monopoly?

“Monopoly” is the only firm (single seller) in its industry.

◮ Question: Is Apple a monopoly? ⋆ It is the only firm that sells iPhone; ⋆ It is not the only firm that sells smartphones.

The formal definition of monopoly is based on the monopoly power. Definition (Monopoly) A firm with monopoly power is referred to as a monopoly or monopolist.

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What is Monopoly Power?

Monopoly power (or market power) is the ability of a firm to affect market prices through its actions. Definition (Monopoly Power) A firm has monopoly power, if and only if

(i) it faces a downward-sloping demand curve for its product, and (ii) it has no supply curve. (i) implies that a monopolist is not perfectly competitive. That is, he is able to set the market price so as to shape the demand. (ii) implies that the market price is a consequence of the monopolist’s actions, rather than a condition that he must react to.

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Profit Maximization Problem

P: the market price that a monopolist chooses; Q D(P): the downward-sloping demand curve that the monopolist faces; Definition (Monopolist’s Profit Maximization Problem) The monopolist’s choice of market price P to maximize his profit (revenue) π(P) P · Q = P · D(P). For simplicity, we assume that there is no production cost, hence profit = revenue. In general, profit = revenue - cost.

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Profit Maximization Problem

The first-order condition: dπ(P) dP = Q + P · dQ dP = 0 The optimality condition: P · △Q Q · △P + 1 = 0

◮ △P is a very small change in price, and △Q is the corresponding

change in demand quantity.

How to understand this?

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Demand Elasticity

Price Elasticity of Demand (defined in Section 3.2.5) η △Q/Q △P/P = P · △Q Q · △P

◮ The ratio between the percentage change of demand and the

percentage change of price.

A Closely Related Question: Under a particular price P and demand Q = D(P), how much should the monopolist lower his price to sell additional △Q units of product? ⇒ Answer: △P = P Q · η · △Q

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Demand Elasticity

The monopolist’s total profit change by selling additional △Q units of product: △π (P + △P) · (Q + △Q) − P · Q = P · Q + P · △Q + △P · Q + △P · △Q − P · Q ≈ P · △Q − |△P| · Q = P · △Q −

• P

Q · η · △Q

• · Q

= P · △Q ·

• 1 − 1

|η|

• ◮ P · △Q is the profit gain that the monopolist achieves, by selling

additional △Q units of product at price P;

◮ |△P| · Q is the profit loss that the monopolist suffers, due to the

decrease of price (by |△P|) for the previous Q units of product.

◮ We ignore the higher order term of △P · △Q. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 25, 2018 12 / 39

Demand Elasticity

Monopolist’s Total Profit Change: △π = P · △Q ·

• 1 − 1

|η|

• ◮ If |η| > 1, then △π > 0. The monopolist has the incentive to decrease

the price.

◮ If |η| < 1, then △π < 0. The monopolist has the incentive to increase

the price.

◮ If |η| = 1, then △π = 0. The monopolist has no incentive to change

the price.

The price under |η| = 1 is the optimal price (if no producing cost)

◮ Equivalent to the previous first-order condition. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 25, 2018 13 / 39

Demand Elasticity

Now consider the production cost, where profit = revenue - cost. Suppose the unit producing cost is C.

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Demand Elasticity

Now consider the production cost, where profit = revenue - cost. Suppose the unit producing cost is C. The optimal price is given by △π = C · △Q, or equivalently, P · △Q ·

• 1 − 1

|η|

• = C · △Q.

Hence at the optimal price, we have |η| = 1 1 − C/P > 1

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Demand Elasticity

Now consider the production cost, where profit = revenue - cost. Suppose the unit producing cost is C. The optimal price is given by △π = C · △Q, or equivalently, P · △Q ·

• 1 − 1

|η|

• = C · △Q.

Hence at the optimal price, we have |η| = 1 1 − C/P > 1 Recall that

◮ When |η| > 1, we say that the demand curve is elastic. ◮ When |η| < 1, we say that the demand curve is inelastic.

Theorem A monopolist always operates on the elastic portion of the demand curve.

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Demand Elasticity

When △Q = 1, then

◮ △π = P ·

• 1 −

1 |η|

• is called the marginal revenue (MR);

◮ C is the marginal cost (MC). In general, MC may not be a constant.

We have MR = P ·

• 1 − 1

|η|

• = C = MC.

◮ The optimal production quality equalizes MR and MC. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 25, 2018 15 / 39

Section 5.2 Theory: Price Discriminations

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What is Price Discrimination?

Price discrimination (or price differentiation) is a pricing strategy where products are transacted at different prices in different markets

• r territories.

Examples:

◮ Charge different prices to the same consumer, e.g., for different units of

products;

◮ Charge uniform but different prices to different groups of consumers for

the same product.

Three types:

◮ First-degree price discrimination ◮ Second-degree price discrimination ◮ Third-degree price discrimination Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 25, 2018 17 / 39

An Illustrative Example

How would the monopolist increase his profit via price discrimination?

◮ MR: the marginal revenue curve; ◮ MC: the marginal cost curve; ◮ Demand: the downward-sloping demand curve;

Quantity Price Q∗ Q⋆ C0 P∗ MC Demand MR π∗ π+ π⋆

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Without Price Discrimination

Without price discrimination, the monopolist charges a single monopoly price to all consumers: The optimal production quality (and demand) is Q∗, which equalizes MC and MR; The optimal monopoly price is P∗, which is determined by the Q∗ and the demand curve. The monopolist’s profit (=revenue - cost) is π∗, and the consumer surplus is π+.

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With Price Discrimination

With price discrimination, the monopolist can charge different prices to different consumers: For example, the monopolist can charge each consumer the most that he would be willing to pay for each product that he buys; With the same demand Q∗, the monopolist’s profit is π∗ + π+, and the consumer surplus is 0; When the demand increases to Q⋆, the monopolist’s profit is π∗ + π+ + π⋆, and the consumer surplus is 0;

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First-Degree Price Discrimination

With the first-degree price discrimination (or perfect price discrimination), the monopolist charges each consumer the most that he would be willing to pay for each product that he buys. The monopolist captures all the market surplus, and the consumer gets zero surplus. It requires that the monopolist knows exactly the maximum price that every consumer is willing to pay for each product, i.e., the full knowledge about every consumer demand curve.

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Illustration of First-Degree Price Discrimination

The consumer is willing to pay a maximum price P1 for the first product, P2 for the second product, and so on. Under the first-degree price discrimination, the consumer is charged by P1 for the first product, P2 for the second product, and so on. The monopolist captures all the market surplus (shadow area). Quantity Price 1 2 3 4 5 6 7 8 P1 P2 P3 P4 P5 C . . . MC Demand

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Second-Degree Price Discrimination

With the second-degree price discrimination (or declining block pricing), the monopolist offers a bundle of prices to each consumer, with different prices for different blocks of units. The second-degree price discrimination can be viewed as a limited version of the first-degree price discrimination (where a different price is set for every different unit). The second-degree price discrimination can be viewed as a generalized version of the monopoly pricing (as it degrades to the monopoly pricing when the number of prices is one).

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Illustration of Second-Degree Price Discrimination

Under this second-degree price discrimination, the monopolist offers a bundle of prices {P1, P∗, P2} with P1 > P∗ > P2.

◮ P1 is the unit price for the first block (the first Q1 units) of products; ◮ P∗ is the unit price for the second block (from Q1 to Q∗) of products; ◮ P2 is the unit price for the third block (from Q∗ to Q2). ◮ The monopolist’s profit is illustrated by the shadow area, and the

consumer surplus is δ1 + δ∗ + δ2.

Quantity Price Q1 Q∗ Q2 P2 P∗ P1 δ1 δ∗ δ2 MC Demand MR

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First- vs. Second-Degree Price Discrimination

Under the second-degree price discrimination {P1, P∗, P2}:

◮ The monopolist’s profit is illustrated by the shadow area, and the

consumer surplus is δ1 + δ∗ + δ2.

Under the first-degree price discrimination:

◮ The monopolist charges a different price D(Q) for each unit of product; ◮ The monopolist captures all the market surplus (the shadow area +

δ1 + δ∗ + δ2, and the consumer achieves zero surplus.

Quantity Price Q1 Q∗ Q2 P2 P∗ P1 δ1 δ∗ δ2 MC Demand MR

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Monopoly Pricing vs. Second-Degree Price Discrimination

Under the second-degree price discrimination {P1, P∗, P2}:

◮ The monopolist’s profit is illustrated by the shadow area, and the

consumer surplus is δ1 + δ∗ + δ2.

Under the monopoly pricing without price discrimination:

◮ The optimal monopoly price is P∗ and the demand is Q∗; ◮ The monopolist’s profit is P∗ · Q∗ −

Q∗ MC(Q)dQ, and the consumer surplus is δ1 + δ∗ + (P1 − P∗) · Q1.

Quantity Price Q1 Q∗ Q2 P2 P∗ P1 δ1 δ∗ δ2 MC Demand MR

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Third-Degree Price Discrimination

Limitation of First- and Second-Degree Price Discriminations

◮ Needs the full or partial demand curve information of every individual

consumer

What happen if the monopolist does not know the detailed demand curve information of each individual consumer, but he knows different total demand curves of different groups of customers? → Third-Degree Price Discrimination

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Third-Degree Price Discrimination

With the third-degree price discrimination (or multi-market price discrimination), the monopolist specifies different prices for different consumer groups

◮ Example: The Disney Park offers different ticket prices to children,

adults, and elders.

Third-degree price discrimination usually occurs when

◮ The monopolist faces multiple identifiably groups of consumers with

different total demand curves;

◮ The monopolist knows the total demand curve of every consumer

group (but not the individual demand curve of each consumer.

How to identify customers?

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By Age

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By Time

Kindle 2

◮ 02/2009: $399 ◮ 07/2009: $299 ◮ 10/2009: $259 ◮ 06/2010: $189 Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 25, 2018 30 / 39

Even More Dynamic

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More Innovative Ways

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Third-Degree Price Discrimination

Consider a simple scenario:

◮ Two markets (groups) of consumers: ◮ The total demand curve in each market i ∈ {1, 2} is Di(P); ◮ The monopolist decides the price Pi for each market i.

How should the monopolist choose prices {P1, P2} to maximize profit?

◮ Whether to charge the same price or different prices in different

markets?

◮ Which market should get the lower price (if the monopolist charges

different prices)?

◮ What is the relationship between the prices of two markets? Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 25, 2018 33 / 39

Third-Degree Price Discrimination

The monopolist’s profit π(P1, P2) under prices {P1, P2} is π(P1, P2) P1 · Q1 + P2 · Q2 − C(Q1 + Q2) The first-order condition: ∂π(P1, P2) ∂Pi = Qi + Pi · dQi dPi − C ′(Q1 + Q2) · dQi dPi = 0, ∀i = 1, 2.

◮ Qi Di(Pi) is the demand curve in market i; ◮ C ′(Q1 + Q2) is the marginal cost (MC) of the monopolist; Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 25, 2018 34 / 39

Third-Degree Price Discrimination

The optimality condition: MC = C ′(Q1 + Q2) = Pi + Qi · dPi dQi = Pi ·

• 1 − 1

|ηi|

• = MRi

◮ ηi Pi

Qi dQi dPi is the price elasticity of demand in market i;

Under the optimal prices (P∗

1, P∗ 2), the marginal revenues (MR) in all

markets are identical, and are equal to the marginal cost (MC): P∗

1 ·

• 1 −

1 |η1|

• = P∗

2 ·

• 1 −

1 |η2|

• Huang & Gao ( c

NCEL) Wireless Network Pricing: Chapter 5 September 25, 2018 35 / 39

Third-Degree Price Discrimination

The optimal prices (P∗

1, P∗ 2) satisfy

P∗

1 ·

• 1 −

1 |η1|

• = P∗

2 ·

• 1 −

1 |η2|

• ◮ If |η1| = |η2|, then P∗

1 = P∗ 2 . That is, the monopolist will charge

different prices when two markets have different price elasticities.

◮ If |η1| > |η2|, then P∗

1 < P∗ 2 . That is, the market with the higher price

elasticity will get a lower optimal price.

◮ Elders will pay less than adults. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 5 September 25, 2018 36 / 39

Third-Degree Price Discrimination

Graphic interpretation of the optimal prices (P∗

1, P∗ 2)

◮ Di: the demand curve in market i; ◮ MRi: the marginal revenue curve in market i; ◮ MR (the blue curve): the overall marginal revenue curve (summing

MR1 and MR2 horizontally);

◮ MC (the red curve): the marginal cost curve;

Quantity Price MC MR D1 D2 MR1 MR2 Q1 + Q2 Q1 Q2 C0 P∗

1

P∗

2

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Third-Degree Price Discrimination

Graphic interpretation of the optimal prices (P∗

1, P∗ 2)

◮ Market 1: the demand is Q1, the marginal revenue equals C0; ◮ Market 2: the demand is Q2, the marginal revenue equals C0; ◮ Total market demand is Q1 + Q2, and the marginal cost is C0; ◮ C0 is at the intersection of MC and MR curves.

Quantity Price MC MR D1 D2 MR1 MR2 Q1 + Q2 Q1 Q2 C0 P∗

1

P∗

2

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Third Degree Price Discrimination

Necessary conditions to make the third-degree price discrimination applicable and profitable:

◮ Monopoly power: The firm must have the monopoly power to affect

market price (there is no price discrimination in perfectly competitive markets).

◮ Market segmentation: The firm must be able to split the market into

different groups of consumers, and also be able to identify the type of each consumer.

◮ Elasticity of demand: The price elasticities of demand in different

markets are different.

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